Appl. Phys. A 70, 475–480 (2000) / Digital Object Identifier (DOI) 10.1007/s003390000437 Applied Physics AMaterialsScience & Processing

A theoretical study on various models for the domain boundariesin epitaxial GaN filmsS.Q. Wang∗, Y.M. Wang, H.Q. Ye

Laboratory of Atomic Imaging of Solids, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110015, P.R. China

Received: 7 April 1999/Accepted: 1 November 1999/Published online: 8 March 2000 – Springer-Verlag 2000

Abstract. Various domain boundaries that are found in epi-taxial Wurtzite GaN films were studied by molecular dy-namics simulation. The Ewald summation algorithm andKeating potential model are adopted to calculate the long-range Coulomb interaction and the short-range bonding forcein the semiconductor system, respectively. The research re-sults show that the domain formation energies of(1100) and(1120) boundaries are significantly different. The latter oneshave general quite higher formation energies than the for-mers. The like-atom (i.e. atoms of the same kind) bondingdomain boundaries (LABDB) have higher formation energiesthan their counterparts of unlike-atom (i.e. atoms of differ-ent kinds) bonding domain boundaries (UABDB) in allGaN(1100) and(1120) interfaces. The UABDB structures are allstable while most of the LABDB are unstable. The advantageand the limitation of Keating potential model in MolecularDynamics simulation for covalent crystal are discussed.

PACS: 61.70; 61.16.D; 68.55.Ln

Gallium nitride (GaN) attracts extensive research interestsboth in science and in industry for it is an important ma-terial in the manufacture of photoelectric device, such asblue/green light-emitting diode, laser diode, etc. The suc-cessful application of the material may cause revolutionarychanges of the human world in illumination and imaging ap-paratuses. A significant character ofGaN is that it can stillgive high-efficiency luminescence even under a high dens-ity of crystal defects. This phenomenon seems in paradoxwith the common opinion that lattice defects introduce non-radiative recombination centres in III-V semiconductor andthe recombination centres reduce the efficiency and lifetimeof the photoelectric devices. The puzzle imposes an urgentdemand for an in-depth study of the defect structures and theirroles in the light-emission process ofGaNcrystal.

∗Corresponding author.(Fax: +86-24/2389-1320, E-mail: sqwang@imr.ac.cn.)

Wurtzite GaN film is usually fabricated by molecularbeam epitaxial method. Because of the lack of suitable sub-strate materials, many crystalline defects form during thefilm growth. The major defects are threading dislocations anddomain boundaries in the epitaxial WurtziteGaN. Thread-ing dislocations have been experimentally identified mainlyto be pure 1/3

⟨1120

⟩edge dislocations, some mixed edge

and screw dislocations and a few〈0001〉 pure screw dislo-cations [1–3]. The exact atomic structure of 1/3

⟨1120

⟩edge

dislocation has been verified by Z-contrast imaging techniquerecently [4]. Theoretical study shows that the edge and theopen-core screw dislocations do not form non-radiative re-combination centres in WurtizeGaN [5], whereas mixed andfull-core screw dislocations give unfavourable effect to the lu-minescent property [4, 6]. Two kinds of domain boundariesrespectively along{1100} and {1120} crystal planes havebeen found inGaN film. Vermaut et al. and Xin et al. pro-posed a double-position boundary model by a defect vectorR= 1/2[1101] for GaN{1120} planar defect [7, 8]. Wang etal. have demonstrated that the like-atom bonding{1120} do-main boundary with defect vectorR= 1/2[1100] can alsoexist in the film [9]. Some others suggested the inversiondomain boundary model of the{1120} defect [10]. Both in-version and double-position domain boundaries by like- orunlike-atom bonding are also suggested forGaN{1100} pla-nar defects [3, 11–15]. From the mechanism ofGaN epi-taxial growth it is understood that the formation of crystaldefect strongly depends on the substrate materials and alsothe growth conditions. Just as proven by experimental ob-servations, the planar defects inGaN film can have variousdifferent structures owing to different substrates and growthconditions.

Northrup et al. studied the surface energy of(1100)and (1120) surfaces, domain formation energy of{1100}inversion, and{1120} double-position domain structures inWurtzite GaN by the first-principles calculations [6, 16, 17].Elsner et al. calculated the electrical properties and line ener-gies ofGaN threading dislocations by ab initio local-densityfunctional cluster approximation [5]. These results have pro-vided useful information for the understanding of lumines-

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cent and structural properties of the material. Quantum me-chanics calculation involves huge mathematical operationsand takes a long time, it can usually only be accomplishedby supercomputers. In another way, one may do the studywith molecular dynamics calculations by using a suitable po-tential model and specially designed computer programs. Al-though the molecular dynamics method is not suitable forthe problems at the electronic level, it may give better resultsfor atomic configurations of defects, impurity diffusion, andphonon properties, etc. by considering a much large system inorder to have a better approach to the reality.

This paper reports our attempt for a molecular dynamicsstudy on the various domain boundary structures in WurtziteGaN. A computer program is developed for the specific cal-culation. For the speciality ofGaN as a covalent systemwith strong ionicity, the long-range interaction between par-ticles is considered by Coulomb potential, and the short-rangeinteraction by Keating potential. The Coulomb potential iscalculated by the Ewald summation method [18, 19]. The po-tential parameters for Keating interaction are deduced fromthe experimental results of elastic stiffness constant [20] byusing a method suggested by Martin [21, 22]. All suggested(1100) and(1120) domain boundary structures by consider-ing both like- and unlike-atom bonds are studied. The domainformation energies and the atom configurations of local lat-tice relaxation for these structures are investigated and com-pared in detail. The result shows that all the LABDBs havemuch higher formation energies than those of their unlike-atom bonding counter-parts. The formation energy of(1100)boundaries is usually lower than that of(1120) boundaries.

The paper is organized to present in Sect. 2 the various do-main boundary models of WurtziteGaN. Section 3 describestechnical aspects for the molecular dynamics calculation. Theresearch results and discussions are given in Sect. 4.

1 Structure models ofGaN domain boundaries

As the result of the extensive studies ofGaN all over theworld in the recent years, various models have been suggestedfor the planar defect in the epitaxial films. It is hard to find anyother materials that have so many structural variations thanthat of GaN domain boundary. The situation partly reflectsthe importance of the defect structure in getting a deep under-standing of the physics property of the material such that somany researchers have been involved to try to find the accu-rate result. It is also partly due to the complexity of the pro-cedure ofGaNepitaxial growth that causes so many structuraldiversities. In the present study, seven typical models forGaN(1100) and(1120) domain boundaries are investigated. Thecorrectness of these models has been proved by experimentalresults. The lattice relationships and atomic configurations ofthese models are given here for the convenience of the presentstudy.

Two different stacking sequences for the atom layers ofboth sides of the boundary have been observed in WurtziteGaN films. The stacks ofN-layers andGa-layers can be insame or inverse orders at the two sides of the boundary. Forthe{1120} domain boundary, there are four typical prismaticstacking fault models proposed to describe its atomic struc-ture. These are two models in the same stacking order and

two models in the inverse stacking order. The lattice config-urations of these models are as follows.

1.1 The(1120) domain boundary models with the samestacking order at the two sides

The operation by cutting aGaN lattice into two parts alonga (1120) plane and shifting a lattice vector ofrl = 1/2[1100]forms a(1120) domain boundary. The atom bonds across theboundary are byGa–Ga, N–N like-atom bonds. For classifi-cation, the model is named as(1120) DBM1. The existenceof this kind of structure in epitaxialGaN was observed inan incipient dislocation, experimentally [9]. On the base of(1120) DBM1 structure, an additional 1/2[0001] shift willeliminate the error atom bounds on the boundary. The newdomain boundary is named as(1120)DBM2. The fault vectoris rl = 1/2[1101]. This kind of structure was initially pro-posed by Drum in his study ofAlN [23], and recently wasfound in GaN film by experiments of high-resolution elec-tron microscopy [7, 8]. The boundary is commonly calledas a double-positioning boundary. The atom configurationsof these two(1120) domain boundaries are illustrated inFig. 1a and b. Besides these twoGaN(1120) domain bound-ary models with the same stacking order at the two sides,a stacking mistake onGaN basal plane can form a different(1120) domain boundary by a fault vector ofrl = 1/6[2203].The boundary structure is just like a deformed DBM2. Due tothe energetic preference [6], the structure will change to the(1120) DBM2 boundary during the film growth.

1.2 The(1120) domain boundary models with inversestacking order at the two sides

These domain boundaries are generally known as inversiondomain boundaries. There are two variations for this class of

Fig. 1a–d.TheGaN(1120) domain boundary models.a andb are the struc-tures of(1120) DBM1 and DBM2 seen along[0001], respectively.c andd are the structures of(1120) DBM3 and DBM4 seen along[1100], respec-tively. The big and small dotsrepresent the two different atoms inGaN,respectively

477

Fig. 2a–c.The GaN (1100) domain boundary models. The(1100) DBM1,DBM2, and DBM3 seen along[1120] are illustrated ina, b, andc, respec-tively

(1120) domain boundaries. The domain boundary by atom-type inverse and a fault vector ofrl = 1/2[1101] is con-structed by like-atom bonds across the boundary. The struc-ture is numbered as(1120) DBM3 in the present study. Byremoving the relative 1/2[0001] shift of the two sub-latticesof the boundary, it forms a boundary of unlike-atom bounds.The boundary is named as(1120) DBM4. The fault vectorof the structure isrl = 1/2[1100]. These domain boundarieswere suggested by Rouviere et al. in theirGaN study [10].The atom configurations of(1120) DBM3, DBM4 domainboundaries are illustrated in Fig. 1c,d.

1.3 The(1100) domain boundary models of WurtziteGaN

Just the same situation as in(1120)domain boundaries, thelayer stacking order can be alike or inverse at two sides ofthe(1100) domain boundary. For the first case, the boundaryis constructed by cutting aGaN lattice into two parts alonga (1100) plane and shifting a fault vector ofrl = 1/2[0001].The boundary is numbered as(1100)DBM1. TheGaNdefectof this configuration had been observed by high-resolutionelectron microscope experiment [15]. For the inversion do-main boundaries on(1100) GaNplanes, the situation is simi-lar to that of(1120) inversion domain boundaries. The(1100)domain boundary that forms by a pure atom-type inver-sion consists of like-atom bonds, and is numbered as(1100)DBM2. Northrup et al. named this structure as a(1100) IDBboundary [17]. Ac/2 shift by fault vectorrl = 1/2[0001]forms a different(1100) domain boundary. It is named(1100)DBM3. Where the atom connections across the boundaryconsist of unlike-atom bounds, the structure is just the(1100)IDB∗ as named by Northrup et al. [17]. Figure 2a–c illustratesall the atomic configurations of those(1100) domain bound-aries discussed in this section.

2 Molecular dynamics calculation

The Coulomb potential plays the main role in an ionic crystal.It is a long-range interaction. A successful energy calcula-tion in an ionic system largely depends on the tactics forhow to deal with the Coulomb force. Besides the long-rangeinteraction, one must also consider the short-range interac-tions between particles in the system. Some potential modelshave been suggested for the short-range interaction, suchas Huggins–Mayer model [24], Tosi–Fumi model [25], andBukingham model [26], etc. The Keating potential model [21,

27] has been proved to be quite successful in the study of elas-tic energy and structural properties of covalent crystals [28–30]. SinceGaN is a strong-ionic covalent crystal, the presentstudy considers the long-range Coulomb interaction by Ewaldsummation calculation, and the short-range bonding interac-tion by Keating potential model.

2.1 The calculation of Coulomb force

The Coulomb potential on the Ith ion is written as

Vi = 1

2

∑j, j 6=i

Zi Zj

r ij.

The effective electric charge of the ions inGaN is takenas Zeff = 1.147, which was calculated from the phononfrequency analysis [31]. For the slowly convergent se-ries of Vi , efficient and accurate calculation of the long-range coulomb potential is quite hard. There are four well-developed methods to realize this kind of calculation, namelythe direct Ewald summation method, the tabulation method,the polynomial-fit-method, and the Fourier transformationmethod [32]. A modified Ewald summation method [18] isused in the present study. To obtain a rapidly convergentresult, the calculation ofVi is separated into three parts asfollows:

Vi = Vig+(Vign+Vi

),

where,Vig is a summation series by putting a like-Gaussiancharge distribution at each ion sitej with total charge ofZj ( j 6= i), Vign is a summation series by putting a like-Gaussian charge distribution at each ion sitej with totalcharge of−Zj ( j 6= i). When the potential of thei th ion isconcerned, the effect of the last two terms in the above equa-tion cancels out each other. The calculatedVig is the totalCoulomb interaction from all the other ions to thei th ion.

2.2 The calculation of short-range interaction in themolecular dynamics study

A Keating potential model is used to calculate the short-rangeinteraction inGaN. The Keating potential is given by

VKeating= 1

2α

(3

4r 20

) 4∑i=1

(ri · ri − r 2

0

)2

+(

3

4r 20

)β

4∑i, j>i

(ri · rj + 1

3r 2

0

)2

,

where,r0 is the equilibrium bond length,ri and rj are bondvectors about the concerned atom.α andβ are the Keatingpotential parameters.

Yamaguchi et al. had measured the elastic coefficients ofWurtizteGaNwith high accuracy. They gave the experimen-tal data in unit of1011 dyn cm−2 as c11= 36.5, c33= 38.1,c12 = 13.5, c13 = 11.4, c44 = 10.9, c66 = 11.5 [20]. Sincethere is not a direct way to obtain the Keating potential pa-rameters from the experimental Wurtzite elastic coefficients,

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these data need to be change to the effective values of its zinc-blende (ZB) counterpart. To get the effective ZB elastic coef-ficients from the Wurtzite data, one can use a least-squares fit-ting calculation. The calculation will give the closest reversetransformation to the known Wurtzite elastic constants. Mar-tin developed a general method for the calculation [22] whichwas modified by Sherwin et al. [33]. By using their methodthe effective ZB elastic coefficients of WurtziteGaNare ob-tained asceff

11 = 33.28, ceff12 = 14.08, ceff

44 = 13.18. Based onthese effective ZB elastic coefficients, it is convenient to getthe effective Keating potential parameters of WurtziteGaNfollowing the way suggested by Martin [21]. TheGaNKeat-ing potential parameters obtained by the above calculationsareα= 97.97943andβ = 20.24743in the unit ofN m−1.

2.3 The computer program of molecular dynamicssimulation

A computer program has been developed to realize the mo-lecular dynamics simulation ofGaN crystal. The simulationis carried out in a canonical ensemble by keeping particlenumberN, system volumeV, and temperatureT constant.The calculation procedures are outlined as follows. A super-cell containingN particles is established at first. Then, toprepare for the molecular dynamics calculation, a Maxwell-distributed initial velocity is assigned to each particle in thesuper-cell. The four tetrahedral-coordinated neighbours arefound for every particle. The following starts molecular dy-namics loops with time interval∆t until the system reachesa stable equilibrium. At the beginning of each loop, theCoulomb and Keating forcesFic and Fik are calculated foreach particle in the super-cell. The total force on the particleis given byFi = Fic+ Fik. The Beeman algorithm is realizedin the particle movement calculation [34]:

ai (t)= Fi /mi ,

vi (t+∆t)= vi (t)+1/6[2ai(t+∆t)

+5ai (t)−ai(t−∆t)]∆t ,

ri (t+∆t)= ri (t)+vi (t)∆t+1/6[4ai(t)−ai(t−∆t)](∆t)2 .

A temperature check is made at the end of each loop to ensurethe simulation is within the limitation of a thermodynamicNTV system.

3 Results and discussions

3.1 The formation energy of the domain boundaries

Periodic boundary conditions were used in the moleculardynamics calculations. There are160 atomsin the super-cells for (1120) and (1100) domain boundary calculations.A super-cell contains two parallel domain boundaries in themiddle and the border. The lattice displacements for theatoms of several lattice periods away from the boundariesremain negligible during the whole course of the simula-tion. Therefore the super-cells in the present study are largeenough to model the isolated domain boundary in the realcrystal. A geometrically optimised lattice configuration isreached after five to ten thousand molecular dynamics steps

in each simulation. The formation energy measures the stabil-ity and feasibility of the domain boundary configuration. It iscalculated byEform= 1/2(E− Ecrystal), where,E is the totalenergy of the super-cell.Ecrystal is the energy of a bulk systemwith an equivalent number of atoms. The domain-wall energyis defined asσwall = Eform/A. A is the area of the domainboundary in the super-cell. In the present studyA= 57.28Å2

for (1120) and A= 33.07Å2 for (1100) domain boundaries,respectively. The calculated results of boundary formation en-ergy and domain-wall energy for the studied boundary struc-tures are listed in Table 1.

From Table 1 it can be seen that the domain-wall energiesof (1120) domain boundaries are usually quite higher thanthose of(1100) domain boundaries. The result may be due tothe fact that there are more serious local lattice distortions atthe(1120) boundaries than at the(1100) boundaries. All thedomain-wall energies of like-atom bonding boundaries arehigher than those of their unlike-atom bonding counterparts.The outcome is caused by the strong repulsive interactionby the same charges. By comparing the calculated domain-wall energiesσwall of (1100) DBM2 and(1100) DBM3 withthat of IDB and IDB∗ in [17], it can be seen that the presentresults forGaN(1100) inversion domain boundaries are qual-itatively consistent with the previous results by the quantummechanics calculations. The quantitative differences of thetwo results are considered owing to the approximations of theempirical potential models in the molecular dynamics calcu-lation. Without any doubt the quantum mechanical methodwill give a more accurate result than that of molecular dynam-ics. The advantage of molecular dynamics is its capabilityto deal with a much larger system for generating qualitativeinformation [35–37]. Table 1 shows that the domain-wall en-ergy of(1120) DBM2 is about three times higher than that of(1100) DBM3, which is similar to Northrup’s conclusion [6].

3.2 The atom configurations ofGaN(1120) and(1100)domain boundaries

The optimised atom configurations ofGaN (1120) and(1100) domain boundaries by molecular dynamics simula-tion are illustrated in Figs. 3 and 4, respectively, where thedark-dot lattices connected by solid lines are the originallattices strictly according to the domain boundary modelsdefinitions as in Sect. 2. The grey-dot lattices connected bydashed lines are the calculated boundary configurations aftermolecular dynamics simulations. The calculated bond lengthsfor theGaN(1120) and(1100) domain boundaries are given

Table 1. Formation energies and domain wall energies forGaN (1120) and(1100) domain boundaries

Domain boundary Eform/eV σwall/meV/Å2

(1120) DBM1 38.81 677.5(1120) DBM2 20.03 349.7(1120) DBM3 32.83 573.2(1120) DBM4 27.47 479.6(1100) DBM1 13.24 400.2(1100) DBM2 10.74 324.8(1100) DBM3 4.45 134.4

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Table 2. The calculated bond lengths for theGaN (1120) and(1100) domain boundaries by molecular dynamics simulations (unit in Å)

Domain boundary r1−2 r1−3 r1−5 r2−4 r2−6 r3−4 r3−7 r4−8 r6−8

(1120) DBM1 1.92 1.74 1.88 1.74 1.88 1.99 2.02 2.02(1120) DBM2 1.86 1.69 1.97 1.69 1.97 1.86 1.97 1.97(1120) DBM3 1.97 1.88 1.94 1.94 2.02 1.97 1.97 1.95(1120) DBM4 1.80 1.76 1.98 1.76 1.98 1.80 1.98 1.98(1100) DBM1 2.13 1.46 1.95 1.53 2.00 2.08 1.99 1.99(1100) DBM2 2.13 1.86 1.92 1.92 2.12 1.92 1.92 1.98(1100) DBM3 1.90 1.78 1.97 1.78 1.97 1.90 1.97 1.97

Fig. 3. The optimized atom configurations ofGaN (1120) domain bound-ary models. Thedark dotsconnected bysolid lines are the original atompositions. Thegray dotsconnected bydashed linesare the optimized atompositions by molecular dynamics calculation

Fig. 4. The optimized atom configurations ofGaN (1100) domain bound-ary models. Thedark dotsconnected bysolid lines are the original atompositions. Thegray dotsconnected bydashed linesare the optimized atompositions by molecular dynamics calculation

in Table 2. The serial numbers of atoms in the table can befound in Figs. 1 and 2. From these results the situation that thestrongly repulsive interactions by like-atoms and the attrac-tive interactions by unlike-atoms at the boundaries is clearlyseen. There are serious asymmetrical lattice distortions at theunlike-atom bonding boundaries. Figures 3a and 4a show thatas the result of lattice relaxation there are irregular arrange-ments for different atoms along the projected direction for theLABDBs. For the UABDBs as in Figs. 3b, 3d, and 4c, the lat-tice distortions are symmetrical and small. The atom columnsare nearly in a straight line along the projection.

3.3 The reliability of the calculation results

In the study of molecular dynamics, how to select a suit-able potential expression is the most important problem.The potential calculation should not be too complex, orelse it is impractical in the simulation. The potential pa-rameters should be directly attainable by using the exist-ing data from experiments. The proper potential should givethe correct lattice configuration by the procedure of mo-lecular dynamics lattice relaxation. Pair potential expres-sions such as L–J potential, Tosi–Fumi potential, etc. haveconsiderable success in the molecular dynamics simulationsin the systems of rare-gas atoms and simple metals. Thestrong direction-preferential interactions in the systems ofsemiconductors impose serious difficulty in using the gen-eral potential calculation methods in the simulation. Thetetrahedral-bonding preference of Keating potential madeit a better approximation to describe the short-range inter-action in WurtziteGaN. The calculation equation is sim-ple, and there are enough experimental data to deduce itspotential parameters. Keating potential can give rather ac-curate description for a small displacement in the system.However, just as Tersoff pointed out [35], the potential be-comes progressively less accurate for large displacementsas the result of itsr 4 characteristic. The time interval ineach step in our simulation is carefully chosen to assurethe maximum atom movement within5% of the equilib-rium bond length. For the three unlike-atom bond bound-aries of(1120) DBM2, (1120) DBM4, (1100) DBM3 andthe like-atom bond boundary of(1100) DBM2, the max-imum bond length changes are less than13.5% during thefull procedure of molecular dynamics simulation. It is con-sidered that these changes are within the valid range ofthe potential, and the obtained optimum lattice configura-tions are good approximations to the real domain bound-ary structures. For the three like-atom bond boundaries of(1120) DBM1, (1120) DBM3, and (1100) DBM1, there isan abrupt change in the total potential energy of the sys-tem at some stage of the simulation. It is explained as theturning point of invalidity of the Keating potential model.This situation happens when the change of bond stretch orcontraction is over20% between the atoms. The atomic con-figurations and domain boundary formation energies for thesemodels are the calculated results by a steps-average beforethe turning point. The real lattice distortions will be moreserious and the formation energies will be higher than thedata listed in Table 2 for these structures, even though the re-sults qualitatively reflect the lattice deformation tendenciesand the structural stableness of the domain boundaries, and

480

can serve as good references for the comparison among dif-ferent structures.

Based on the theoretical results by the present study, itcan be concluded that there will be more(1100) rather than(1120) domain boundaries in WurtziteGaN from the ener-getic viewpoint. This is in accordance with the usual experi-mental observations that most grain boundaries are(1100)boundaries. All the unlike-atom bonding domain boundariescan form stable atom configurations. The atom positions areregular and symmetrical at the boundaries. Most of the like-atom bonding domain boundaries are unstable. The atompositions are irregular and lack symmetry at the boundaries.A simple way to differentiate between these domain bound-aries can be that the atomic images by unlike-atom bound-aries will be clearly distinguishable regularly arranged imagepatterns, whereas the images of like-atom boundaries willcontain many irregular patterns and be seriously blurred inhigh-resolution electron microscopy. The like-atom bonding(1100) DBM2 boundary is an exception. Its domain forma-tion energy is not too high. The boundary configuration is sta-ble and there is no serious lattice distortion. It can be verifiedby contrast difference across the boundary by high-resolutionelectron microscopy.

4 Conclusions

A molecular dynamics study on various domain boundarymodels in epitaxial WurtizeGaN film is accomplished byusing a specially designed computer program. The long- andshort-range interactions in the crystal are handled by Ewaldsummation and Keating potential calculation, respectively.The present study shows that although the Keating potentialexpression has serious faults for large-range bond change, itis able to give reasonable result when the change is small.The (1120) domain boundaries have generally higher for-mation energies than(1100) domain boundaries in WurtziteGaN. Therefore the number of(1120) domain boundariesis less than the number of(1100) domain boundaries in thematerial, based on the same growth condition. The forma-tion energies of like-atom boundaries are all higher than theirunlike-atom counterparts. A simple structural differentiationof these boundaries can be made by high-resolution electronmicroscopy.

Acknowledgements.This work is financially supported by the National Pan-deng Research Project of China under the grant number of 95-Yu-41.

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